The reason why the picture "proof" 3=1 is wrong

in #mathematics2 years ago (edited)

Hi Steemians! This is continuation of the picture "proof" that 3=1 post. In this post I will explain why the picture "proof" is wrong.

A simple example

I will first treat an example where the problem is more evident. Take this curve:


Its a which is formed out of two straight lines of length 1 which are placed at an angle of 90 degrees. If I change the angle a bit its total length will still remain the same. However, if I change the angle by 90 degrees this happens:


So the total length of the blue curve becomes 1. This does not imply that 2=1 because in a sense the line has been folded.

Returning to the 3=1 picture "proof"

Recall these pictures:


Also recall that the green line has length 1. I will assign a different color to the horizontal lines of the baby blue curve and I will also assign two different colors to the vertical lines of the baby blue curve:


I will collect all the orange lines in one line and I will decrease the angle between the pink and purple lines with each iteration:


The inclined angles will become smaller and smaller. Imagine that I would continue this procedure by taking the limit to infinity. Then you will see that I will cover the green line not one time but three times. In some sense it is very similar to the earlier example.


This "proof" shows that length is a relative concept. Suppose you fold a piece of paper then somebody might say that its length has been halved while somebody else might say that its length remains the same. Who is right? That can only be determined once you have determined what the right way is to measure a folded piece of paper. Then you know who is right. But you know for sure that both cannot be right at the same time.


Figures made with inkscape. It is free!

Thank you

Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below. :o)

Owl tax

Photo by Ciggy from the Pixabay - CC0


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